Optimal. Leaf size=34 \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]
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Rubi [A] time = 0.0070304, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {364} \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^n\right )^2} \, dx &=-\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0035451, size = 31, normalized size = 0.91 \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (n + 1\right )} \int \frac{1}{a b n x^{2} x^{n} + a^{2} n x^{2}}\,{d x} + \frac{1}{a b n x x^{n} + a^{2} n x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{2} x^{2 \, n} + 2 \, a b x^{2} x^{n} + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.39887, size = 289, normalized size = 8.5 \begin{align*} - \frac{n \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{n \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{\Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{b n x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a^{2} \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{b x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a^{2} \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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